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Irreducible representations of the symmetric group Sym$(n)$, and degree-$n$ algebraic representations of SL$_d(\mathbb C)$ for $d\ge n$, can both be classified by Young diagrams with $n$ boxes.

Consider now a degree-$n$ irreducible representation of SL$_d(\mathbb C)$ with $d\ge n$, and restrict it to the subgroup of permutation matrices Sym$(d)$. What is its decomposition in irreducibles? Is there a nice way of listing those Young diagrams (with $d$ boxes) which occur, constructed out of the Young diagram with $n$ boxes of the original representation?

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up vote 6 down vote accepted

This is given in Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the restrictions in terms of plethysm and inner product.

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Thanks a lot ! – grok Apr 7 '13 at 9:37

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